mathematics model: safety estimation of keep to right rule at freeways

For office use only

T1 ________________

T2 ________________

T3 ________________

T4 ________________

Team Control Number

31234 Problem Chosen

A

For office use only

F1 ________________

F2 ________________

F3 ________________

F4 ________________

2014 Mathematical Contest in Modeling (MCM) Summary Sheet

In this paper, we try to simulate and analyze "keep right except to pass rule in

USA traffics. We want to generate the crash probability by applying the rule to

various situations in freeways, such as traffic density, distance between car, and even

bad weather. By changing these factors, we examine the effectiveness the rule with

various traffic conditions.

For simulating and analyze this rule, we apply two basic formulas and several

assumptions, including homogenous vehicle assumption of microscopic model and

one way direction of freeway. The based theories which we use are General Motor

Model at Traffic Flow and Indicator Probability Function. The first step to attain our

model is applying General Motor model with some algebraic modifications to allow

us predict the least response time needed to pass the front car before collision occurs.

After we find out the response time, we develop crash indicator function as the

comparison tool to decide which situation is better to pass.

We also take into consideration the effect of Intelligent System in automobile

by calibrating model regarding its external factors. By eliminating human error at our

model, we get the decrement of crash probability. In brief, we conclude that our

model is quite plausible to analyze the effect of external factors.

We apply our model to validate traffic criterion on Highway Capacity Manual

published by Transportation Research Board. At this study case, we find that our

model can reconfirm the precise criterion of traffic condition, including its safe

distance. We believe our model can be applied to make best traffic criterion and rule

with different circumstances.

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Table of Contents

Introduction ...........................................................................................................2

Traffic Flow Theory ..............................................................................................2

Macroscopic Point of View .......................................................................3

Microscopic Point of View........................................................................4

Indicator Function Theory ...................................................................................4

General Assumption .............................................................................................5

Description of Model.............................................................................................5

Flowchart ...................................................................................................5

List of Variables ........................................................................................6

Model Construction ..................................................................................6

Analysis of Model ..................................................................................................11

Sensitivity and Stability Analysis ........................................................................13

Testing Model ........................................................................................................14

Strength and Weakness ........................................................................................16

General Solution and Improvement ....................................................................17

Conclusion and Recommendations .....................................................................18

References ..............................................................................................................19

Appendix ................................................................................................................20

List of Figures

Figure 1: Work process to general solution .............................................................. 5 Figure 2: illustration to change lane to left ............................................................... 6

Figure 3: illustration to change lane to right lane ..................................................... 9 Figure 4: Comparison between Poisson Cumulative Distribution and our Indicator

Function .................................................................................................................. 11

Figure 5: 3d Graphs of Probability of Crash with speed and distance as input ...... 13 Figure 6: Categorization of LOS by Transportation Research Board of the National

Academies of Science in United States .................................................................. 15


INCREASING SAFETY TO PASS

BY USING CRASH INDICATOR FUNCTION

Introduction

Freeway or sometimes called controlled access highway is a high speed design

of road with ramps to regulate its ingress/egress. The development and creation of

freeway have many effects on America, including its safety issue. There is no doubt

that freeway give many benefits such as increment of economic productivity and

improvement national competitiveness of America. On the other hand, freeway has its

own risk of crash, even though the keep right lane except to pass rule is

implemented.

The rule, actually, is an ambiguous one and varies between states. It is not

exactly tells the driver not to use left lane unless the driver want to pass front car.

However, we decide to strengthen the rule by explicitly state that the driver has to

move on lane to the left and back to his/her previous track after passing another

vehicle. The main purpose of this rule is to decrease the congestion when peak time.

In this paper, we will assess whether this rule would increase the chance of crash

between cars using simple trajectory assumption.

The primary objective of this paper is to give a relation between crash

probability with various velocities and distances, due to different traffic density, of the

car which wants to pass another vehicle using the rule. Another objective in this paper

is to decide whether the model can be carried over to other countries which driving on

the left is the norm and later we will consider the effect of intelligent system in the

rule. In this paper, definition of crash is whenever a part of car is collided with part of

another vehicle. To achieve this linkage, some background theories are needed, such

as Indicator Function Theory and Traffic Flow Theory.

Traffic Flow Theory

Macroscopic Point of View

Traffic Flow Theory has an important part of our paper because it gives us

basic model and many assumptions to develop furthermore. In this paper, our main of


model core is Microscopic Traffic Flow Model, even though the macroscopic theory

is still needed to explain varying situation of traffic. Here are several variables and

concepts which are used to categorize state of traffic.

Density

Density, which is denoted by , reflects the number of vehicles per km of roads.

The index n indicates number of vehicles at exact time within the L length freeway.

Intuitively, density interprets the crowd on the road at exact time. In traffic flow, there

are two important types of density, which are critical density ( ) and jam density

( ). Critical density occurs under free flow condition. On the other hand, jam density

is achievable under congestion. Inverse of density formula is called as average

distance between cars ( ).

Flow rate

Flow rate or flow ( ) is a number of vehicle passing reference point per unit time

( ). Usually, flow rate is written by number of vehicle per hour. The concept of

traffic flow rate is quite same with flow rate of fluid theorem, which is the discharge

or flux of the stream.

In this paper, we categorize traffic by its density. By changing the traffic flow

density input and minimum velocity of the freeway, we would like to analyze the

change of crash chance of car when passing other car. The traffic categorization we

use in this paper is based on the Level of Service which is published at Highway

Capacity Manual. Level of Service measures quality of service of the freeway.

Highway Capacity Manual describes LOS as a traffic conditions within a traffic

stream. On the freeway road, LOS is measured based on traffic density


Microscopic Point of View

As we stated above, our approaching of this model is based on microscopic

traffic model. We use microscopic traffic model because of its simplicity as the model

just consider interaction between two cars in the same lane. Microscopic model also

consider driver behavior in following the other car from the same lane. We account

car following model as our based model in this paper. There are three interesting car

following model: Pipes model, Forbes model, and General Motors model (GM

model). In this paper, we decide to use General Motor model because of its agreement

at field data and synchronization with macroscopic model.

Basic philosophy of this GM Model is from Newtonian Mechanics, which is

account acceleration as a response of stimulus it receives in form of the force as an

interaction to another particle in system. However, in this paper, we try different

approach based on this model which assumes response time or stimulus is composed

by velocity of vehicle, relative velocity between vehicle, and distance. The

cornerstone of GM Model itself is Follow the Leader model. Therefore, GM Model

and our model in this paper, follow two assumptions from Follow the Leader model:

Higher the speed (velocity) of the vehicle, higher the distance between the

vehicle.

To avoid collision, driver has to maintain safe distance with vehicle ahead.

In General, our based model has form

, denotes safe distance between two car either at left lane or right lane,

, denotes response time ( ) times different velocity between vehicle on

same lane.

Indicator Function Theory

Indicator function or a characteristic function is a function, defined on set

which maps every member of set to closed interval of real number between 0 and

1. By mathematics notation,


In this paper, we use indicator function to describe the probability of crash when a

vehicle wants to pass another vehicle ahead.

General Assumptions

Due to large factors related to crash at freeway, we make several assumptions to

simplify our problem to determine crash probability in the freeway that has keep

right unless to pass rule. Here are the assumptions:

Creation of moving frame which is caused by microscopic model. The frame

will move with same velocity at average right lane velocity. By using this

assumption, we take into account 3 vehicles on the frame. The first vehicle is

slow vehicle on the right, second vehicle is faster one that may to pass,

and last vehicle is disturbance vehicle on the next lane.

Velocity of vehicles on the same lane is same unless it wants to pass another

vehicle ahead.

Vehicle on left lane is faster than vehicle on right lane. For basic model, we

take consider two lane freeway with same direction.

The freeway just has one direction and has no bend.

We do not consider length either width of the vehicle.

The distance between vehicles on same lane is always same.

Description of Model

Flowchart

Before we start to derive our model, we would like to give a brief explanation

about the whole process to get general solution. Here is our work diagram,

Figure 1: Work process to general solution


List of Variables

= speed of the passing car ( )

= speed of the passed car ( )

= speed of the next lane car( )

= proper angle to move successfully to next lane ( )

= time needed to move to next lane ( )

= length between the center of lane (width of the lane)( )

= safe distance between two respective cars on left lane ( )

= safe distance between two respective cars on right lane ( )

= exact distance between -car and -car in the same lane ( )

= response time ( )

Model Construction

The figure 2 represents a frame of

two lanes freeway with three cars. We

focus on the -car (this car has constant

speed ) which has willingness to pass

the -car (this car has constant speed .

Then on the left lane there is -car (this

car has constant speed ) running without

considering anything on the right lane. As

the our assumption hold, the average speed

of vehicles in left lane should be faster

than the right lane one. So, we can easily

state that our frame obeys .

Now, we let -car is going to pass the front car and the car must move to left lane

and then move again to the former-right lane. Of course before -car move to the

left lane, it should consider the running -car beside. If the -car position is still

behind the -car, of course -car will have bigger successful chance to move to

left lane safely. But, we are interested in something more threatening, that is the -

car will start to move to left lane when the -car is exactly beside the -car. This

requirement is good enough in testing our passing rule even in our other frames.

Figure 2: illustration to change lane to left


Notice that when -car is going to move to left lane, it has to choose proper

angle in order to keep a distance at least with -car. So now we are going to

find the relation of angle and time needed to move to left lane, that is .

First, there are two ways to describe , those are by using trigonometric ratio

from or by considering the length of car should run to the left lane, .

Mathematically, we can write the equation as

By equating the second and the fourth term of the equation, we get result

,

as we desired before.

Meanwhile the car on left lane, -car, will run for time and speed so we

directly know that After the passing car, -car, has arrived on left lane

and be in the front of the -car, that car should give a minimal gap (safe distance on

left lane) which we notate as in order to avoid the collision between those cars.

So

We substitute and to last inequality and we have

Now, substitute

again and we will obtain good inequality with only 1

variable

Set

and use the trigonometric formula to the left side will make it become

simpler


with or

Finishing the last inequality we will have a new inequality

(

)

If we take the equality of the last inequality, it means the least angle (in radian) that

the passing car should take to have a successful move to left lane without crashing the

-car.

Now we should consider about the distance between the passing -car and

the -car. First, we define is minimum time for -car to collide the -car

from the rear because and it can be notated as

with is the exact distance between -car at that time and -car. Of course

because is the least distance (minimum safe distance) in order to

avoid the collision on right lane.

Because the -car move linearly to the left lane, so the time for moving the

left lane should be less than of or .

After -car has satisfied requirement, we define as the response

(sensitivity) time to have -car overcome the threat of colliding the car(s) on the

same lane. Considering there is only -car on the lane, we formulate the relation

between -car and -car as

We wish for as function of , therefore

Our model has a requirement that the higher average speed of vehicles on a

lane then the higher safe distance between two respective vehicles. Since the

average speed on left lane is strictly higher than the right lane, we conclude that

. Relying on this, we obtain that


Then remember that and the time should fulfil the

requirement

. So then

and apply this

to right hand of the inequality

Or

The value of is still varying depends on , but we can choose to obtain

the , that is

If we take the angle exactly when (

) with

, the -car will be exactly on the left side of -car when it has arrived on

left lane, as in the figure 3.

When the -car has a successful move

to left lane, it should consider how to go back to

right lane as the driver should obey the passing

rule. This time, the -car can be considered as

the -car before when -car moved to the left.

But we should note the important difference that

. So, logically when the -car is

moving back to the right lane again by the same

angle when it moved to the left lane, it will

have higher chance to move successfully. So we

focus only on collision threat from -car in the

front.

If we recalculating the time needed to move to right lane and the reaction

time with the configuration as in the figure 4, it has look-liked results as we

obtained before i.e.

Figure 3: illustration to change lane to right lane

Team # 31234 Page 10 of 21

( )

( )

Note that the absolute value comes from changing the direction.

Because the -car is on the left lane, it must follow that . We are

interesting in finding the minimum reaction time , choose and we will get

as we deserved as

( )

( )

Finally, we are ready in constructing the model for measuring the crash rate

depends on response time . Remembering our assumption that we always focus on

the frame of -car behavior, we can relate in our frame to a parameter of Poisson

distribution. Furthermore, what we analyze about the crash rate is how long a driver

needs to avoid the collision at seconds or lower. Thus, crash rate is indicated by the

Indicator Function as written as

with is the external factor coefficient which . If is getting near to , it

means that the external factor is nearly none; otherwise external factor is very

influencing the driver. In our model, the external factor is only caused by weather,

drunk driver, and drivers distraction; and three of them have their own percentage.

By data from statistics about how big the relation between our external coefficients

and safety while driving, we have a formulation

with

and define for if the corresponding factor exist(s); and for

otherwise.

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Analysis of Model

In our model, we are heavily relied on response time of the driver with various

circumstances. According to that reason, we need to find average response time on

driving. In this paper, we use 3 seconds as average response time based on

National Safety Council of USA. In accordance to definition of response time, we

conclude that if , the driver will have less probability to crash with

another vehicle.

We choose a function, which is called Indicator Function, to give a

quantitative result of crash probability based on response time ( ). Indicator Function

has an ability to give a fair comparison for different input of response time because it

has codomain of closed interval [0,1]. Our model of crash indicator function has a

good relation with the CDF of Poisson distribution because the role of in our

assumption about moving frame inspired from how Poisson distribution can be

applied to a random variable. It seems delicate since we focus only on how big the

crash probability when the driver has reaction time or lower. By plotting graph of

Cumulative Distribution Function of Poisson with and our indicator function,

we find that more likely response time to happen, less probable crash to occur.

Speed and distance are the main factors in evaluating response time ( ), as

well as main factors of evaluating the crash probability. However, we try not to

neglect any other factors. Finally we consider some external factors such as bad

weather, drivers distraction, and drunk driving. These three factors can be formulated

in a coefficient called . Of course, this coefficient affects our work on crash indicator

function. Value of c is a real number, ranging from zero to one ). Other

property of is greater the value of c, less chance of crash to occur.

Finally we state our model of Crash Indicator Function as

Figure 4: Comparison between Poisson Cumulative Distribution and our Indicator Function

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We now test the model with chosen condition. Giving the condition of the

traffic, we obtain table as stated below.

Traffic condition: , , Traffic condition : , ,

27 5 0.7621 10.0239 1.1124 0.8685 29 5 0.7097 10.0223 1.1123 0.8685

35 0.191 40.006 7.7784 0.0083 35 0.1778 40.0056 7.7783 0.0083

70 0.1019 75.0032 15.5559 0 70 0.0948 75.003 15.5559 0

105 0.0694 110.0022 23.3336 0 105 0.0647 110.002 23.3335 0

140 0.0527 145.0017 31.1113 0 140 0.0491 145.0015 31.1113 0

29 5 2.1201 3.3555 0.404 0.9306 31 5 1.9841 3.3541 0.4037 0.9306

35 0.5332 13.3389 2.8018 0.5494 35 0.4988 13.3386 2.8016 0.5494

70 0.2845 25.003 5.601 0.0691 70 0.2661 25.0028 5.6009 0.0691

105 0.194 36.6687 8.4007 0.0045 105 0.1815 36.6686 8.4007 0.0045

140 0.1472 48.3349 11.2005 0.0003 140 0.1377 48.3348 11.2005 0.0003

31 5 3.2946 2.0207 0.2093 0.9422 33 5 3.0966 2.0195 0.209 0.9422

35 0.8312 8.0052 1.4309 0.8277 35 0.7809 8.0049 1.4308 0.8277

70 0.4435 15.0028 2.8585 0.5353 70 0.4166 15.0026 2.8584 0.5353

105 0.3024 22.0019 4.2866 0.2164 105 0.2841 22.0018 4.2866 0.2164

140 0.2294 29.0014 5.715 0.0621 140 0.2155 29.0014 5.715 0.0621

33 5 4.3208 1.448 0.1292 0.9464

35 1.093 5.7192 0.8668 0.8941

70 0.5832 10.7169 1.7299 0.7808

105 0.3977 15.7161 2.5936 0.6002

140 0.3017 20.7156 3.4576 0.3876

Traffic condition: , , Traffic condition: , ,

31 5 0.664 10.0209 1.1123 0.8685 33 5 0.6238 10.0196 1.1122 0.8685

35 0.1663 40.0052 7.7783 0.0083 35 0.1562 40.0049 7.7783 0.0083

70 0.0887 75.0028 15.5558 0 70 0.0833 75.0026 15.5558 0

105 0.0605 110.0019 23.3335 0 105 0.0568 110.0018 23.3335 0

140 0.0459 145.0014 31.1113 0 140 0.0431 145.0014 31.1113 0

33 5 1.8645 3.3529 0.4035 0.9306

35 0.4686 13.3382 2.8015 0.5495

70 0.25 25.0026 5.6009 0.0691

105 0.1705 36.6685 8.4006 0.0045

140 0.1293 48.3347 11.2005 0.0003

With varying the condition of traffic flow, we have obtained the crash

probability depend on various speed and distance between vehicle. By analyzing the

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table, we may conclude that as the distance increase and speed decrease, less probable

crash to occur. For giving more insight about how our model behave with different

input (speed and distance), we give 3d plot using Maple.

Graphs of our crash probability model using Maple (fixed parameter, vn= 30m/s and

vp = 32 m/s)

Figure 5: 3d Graphs of Probability of Crash with speed and distance as input

Two graphs above give a visualize behavior of our model with various speed

and distance. For both of the graphs, range of speed that we use is from 32 to 39 while

the distance is ranging from 5 to 150 meters. Our model is quite consistent with our

based model as from the graph can be concluded that crash probability is

proportionate with speed and inversely proportional to distance.

Sensitivity and Stability Analysis

The model that we create has a rule to pass vehicle by changing to the left

lane. Because there are some countries which have the exact opposite orientation rule,

that is the keep left except to pass rule, we can create a new model only by changing

the orientation of variable. The result of our new model will be exactly the same.

However, from some articles that we found, changing to the left traffic rule

give more accident rates than changing to the right traffic rule. Our model cannot

explain why this happen. By some researches, we found that the accident rate is more

affected by the drivers habit and human dominant eye and hand. Most of human can

use their right hand better than their left. For example, by changing orientation, people

who used to grab shift gear with his left hand (left hand rule) can adjust better than

people who have to grab shift gear with his left hand. Our model do not consider these

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variable, we believe by adjust oneself to different rule, the probability of crash will

decreases as the time flies.

Another case we want to consider is the effect of external sources; our model has

considered these treatments in constant c. We will analyze the effect of constant c

with some number.

Here is the initial condition

vn vp vn1 dx teta t

24 26 29 70 0.2845 25.003 5.601

Notes:

v: the external factor is

present on the moving frame

We obtain the presence of external factors is significant to the crash

probability. Consider when there is none of these factors present, the crash probability

is below than 10%, compared to all external factors are present, the probability

increase to more than 50%. With the increment of external factor, higher crash

probability will be obtained.

If the vehicle is under control of intelligent system, external factor like drunk

driving and distraction can be ceased. It implies that the weather alone could affect

crash probability and it has less crash probability comparing to human error effects.

Furthermore, the existence of intelligent system makes the speed and turning

angle, are more precise. We conclude that the existence of intelligent system makes

our model more accurate and, generally, decrease the probability of crash.

Testing Model

For test case, we decide to choose USA traffic categories from Highway

Capacity Manual, LOS criteria. Here is the table from Highway Capacity Manual

about LOS category,

Drunk Driving Distraction Weather c P(crash)

1 0.0691

v 0.68 0.3082

v 0.84 0.1538

v 0.89 0.1208

v v 0.5712 0.4503

v v 0.7476 0.2337

v v 0.6052 0.4038

v v v 0.5084 0.5381

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We are interested to check the crash probability of this system with our model.

The condition in the moving frame is set to LOS A, LOS B, LOS C, LOS D, and LOS

E. We take the lowest minimum speed 24.59 (55mph) as the speed of vehicle in

the most right lane and vehicle on next lane has average speed of 26.82 (60mph).

Suppose there is a vehicle which is running at 29.06 (65 mph) enter the moving

frame, we will give a crash probability of every type of LOS.

Criteria dx

LOS A 146.3 24.59 26.82 29.06 0.1087 65.314 16.3284 0

LOS B 89.41 24.59 26.82 29.06 0.1778 39.9177 9.9794 0.0009

LOS C 61.9 24.59 26.82 29.06 0.2568 27.6375 6.9094 0.0197

LOS D 45.98 22.45 25.85 29.06 0.4954 14.3288 3.0313 0.4922

LOS E 35.76 22.35 22.84 29.06 1.2333 5.7554 0.3945 0.9312

Figure 6: Categorization of LOS by Transportation Research Board of the National Academies of Science in United States

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In LOS A, the crash probability is nearly zero, or we assume it is safe for the vehicle

to run at 60 mph and pass another vehicle ahead.

In LOS B, the crash probability is 0.0009 or we can interpret for every 1000 cars

which is running at 60 mph, there will be nine cars, approximately, get into accident.

In LOS C, the crash probability is increasing to nearly 2%, a quite acceptable number

but still, it is a warning for every driver who wants to pass.

In LOS D and LOS E, the probability is really high around 50% and more than 50%,

we consider this number really high. In order to pass another vehicle safely in LOS D

or LOS E the vehicle must reduce its speed, or the collision will likely to happen.

If we consider LOS A to LOS E as light traffic to heavy traffic, it says that faster the

vehicles, easier to have a crash in traffic flow. Looking at his test case, we can

conclude that LOS categorization which is published on Highway Capacity Manual

give information to the driver about maximum speed to avoid collision.

Strength and Weakness

Model Strength

The model is simple enough to represent the problem of passing another

vehicle. User can only input the speed of vehicles which are in the frame and

input safe distance. With only those steps, we will know whether the rule of

passing is good or not by looking at the crash probability.

Model Weakness

The model does not consider that the vehicle can accelerate or decelerate since

it often occurs on road.

Although we already have the coefficient c as the external factors (drunk

driving, drivers distraction, and bad weather), it still not occupy all the

external factors in reality. This means that the coefficient c still needs to be

revised more.

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General Solution and Improvement

When the -car try to pass the -car, the driver must consider about the

crash probability two times. First is when moving to left lane and second is when it

tries to move back to former-right lane. Both of the crash probabilities are different, of

course, but we should have a single value crash probability of passing, directly. This

is important in order to know that this rule is good or not. Moreover, we can compare

it with another rule.

As we discussed before about the -car passing the -car by applying the

keep-right-most-lane-unless-passing rule, the car should success when move to left

lane first. This means that we at least need the probability as to success

move to left lane and continue its move. Regarding the last move back to right lane,

we formulate the crash joint probability as

( )

Now, we try to apply the reverse of keep-right-most-lane-unless-passing rule,

lets say it as keep-left-most-lane-unless-passing rule which is implemented in some

countries such as Indonesia, UK, Australia, etc. Fortunately, our model still fits with

this reversed-rule without changing any property of our model. Note that if the -

car want to pass the -car, it should move to right lane first and then going back to

former-left lane. Therefore if we define as the crash probability of moving

to right lane with our reversed rule, this follows . It will be

the same if we define as the crash probability of moving back to left lane

then .

Finally, we obtain the crash joint probability of applying the reversed rule as

(

) ( )

Thereafter, by using our model, we conclude that using whether the keep-right-most-

lane-unless-passing rule or keep-left-most-lane-unless-passing rule has no different in

comparing the crash rate.

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Conclusion and Recommendations

The longer safe distance between two respective vehicles on same lane, the

longer response time needed.

The further difference between the speed of passing car and the speed of

passed car, the faster response time needed.

Response time of drivers in USA has average of 3 seconds. This means that if

the response time is further below from 3 seconds, then the crash probability

will increase exponentially. Otherwise, if the response time is further above of

3 seconds, then the crash probability will decrease exponentially.

Through our model, both of left passing rule and right passing rule have the

same rate of crash. In other word, both rules have no difference significantly

to driving safety.

Our model still fits on using the autonomous vehicle after neglecting the

drivers error in driving.

For the future research with our approach and model, there are some

recommendations:

Start to consider acceleration or deceleration of all vehicle especially the

passing vehicle

When the passing car have to move to other lane, it is better to use curve

trajectory instead of linear trajectory which we use in our model. Curve

trajectory is more possible to happen in real road.

The crash indicator function should consider more other external factors of

safety driving. This is important in order to enhance the power of indicator

function as comparison tool of some traffic rules.

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References

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Basacik, D and A. Stevens. Road Safety Research Report No. 95 Scoping Study of

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Immers, L.H. and S. Logghe. Traffic Flow Theory. Katholieke Universiteit Leuven.

Belgium. 2002

McManus, I.C et. al. Eye-dominance, Writing Hand, and Throwing Hand. University

College London, UK University of Victoria, Canada University of Waterloo,

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Oh, Cheol and Taejin Kim. Estimation of Rear-End Crash Potential Using Vehicle

Trajectory Data. Department of Transportation Systems Engineering, Hanyang

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Transportation Research Board. Highway Capacity Manual 2000. National Research

Council. United States of America. 2000

Driver Reaction Times in Road Traffic,

http://hugemann.de/pdf/evu_2002_reaction_german.pdf

Driving Defensively,

http://www.nsc.org/news_resources/Resources/Documents/Driving_Defensive

ly.pdf

Indicator Functions, http://www.statlect.com/indica1.htm

Microscopic Traffic Flow Modeling,

http://www.cdeep.iitb.ac.in/nptel/Civil%20Engineering/Transportation%20En

gg%201/34-Ltexhtml/nptel_ceTEI_L34.pdf

Reducing the Legal Blood Alcohol Concentration for Driving in New Zealand,

http://www.ahw.org.nz/resources/pdf/AHWbriefingpaper_2003-1_BAC.pdf

http://www.statisticbrain.com/car-crash-fatality-statistics-2/

http://en.wikipedia.org/wiki/Dagen_H

http://en.wikipedia.org/wiki/730_(transport)

http://safety.fhwa.dot.gov/geometric/pubs/mitigationstrategies/chapter3/3_lanewidth.h

tm

http://en.wikipedia.org/wiki/Blood_alcohol_content

Team # 31234 Page 20 of 21

Appendix

Code Printout

For doing some calculation and plotting graph on this paper, we use both of Maple

and MatLab software. Here are its source codes

Matlab Source Code

clear all;

close all;

clc;

vn=24;

vn1=29;

vp=26;

% dxleft=[10 40 75 110 145]; //for testing various traffic condition

dxleft=75; //for testing external factor

% dxright=[5 35 70 105 140]; //for testing various traffic condition

dxright=70; //for testing external factor

% dx=70 //fpr testing external factor

r=3.6;

teta=atan(-dxleft/r)+acos(vp/(vn1*sqrt(1+dxleft^2/r^2)));

teta/pi*180 //convert radian to degree

t=r/(vn1*sin(teta))

tao=((dxright*(vn1-vn)-dxright*(vn1*cos(teta)-vp))/(vn1-vn)^2); //count reaction

time

a1=1; //constant for drunk-driving

a2=1; //constant for distraction

a3=1; //constant for weather

c=(1-a1*0.32)*(1-a2*0.16)*(1-a3*0.11) //c formula

i=1/(1+exp(c*tao-3)) //indicator function

% Right to Left //for various traffic condition

% j=1;

% while j

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Maple Source Code

//restart; with(plots)

//vn := 30

//vp := 32

//r := 3.6

//plot3d(1/(1+exp(((dxleft+5)*(vn1-vn)-(dxleft+5)*(vn1*cos(arctan(-

dxleft/r)+arccos(vp/(vn1*sqrt(1+dxleft^2/r^2))))-vp))/(vn1-vn)^2-3)),

vn1 = 32 .. 39, dxleft = 5 .. 150, axes = normal)

//plot(1/(1+exp(tao-3)), tao = 0 .. 35)

mathematics model: safety estimation of keep to right rule at freeways

Documents

description of model

model construction

analysis of model

general motor model

traffic density

best traffic criterion

traffic flow theory

various traffic conditions